Probability of landing inside the convex hull of previously sampled points

Let $$\{X_i\}_{0\leq i\leq\infty}$$ be i.i.d. random vectors in $$\mathbb{R^d}$$. I would like to show that the probability of one point being in the convex hull of the others goes to one with the number of points: $$\tag{1}\label{claim} \lim_{n\to\infty}\mathbb{P}(X_0\in\mathrm{Conv}(X_1, \ldots, X_n)) = 1,$$ where the probability is taken with respect to the realization of $$(X_0, \ldots, X_n)$$.

While this seems intuitive, the following is a counterexample: in $$\mathbb{R}^2$$, if the $$X_i$$ are uniformly sampled from the circle $$\mathbb{S}^1$$, every sampled point almost surely creates a new vertex of the convex hull, and therefore $$\mathbb{P}(X_0\in\mathrm{Conv}(X_1, \ldots, X_n)) = 0$$.

If the $$(X_i)$$ are sampled from a continuous distribution, is my initial claim $$\eqref{claim}$$ true?

• In your counterexample the distribution is not discrete. What do you mean by "continuous" distribution? Aug 24 '20 at 23:55
• This question is somewhat relevant, as it shows that it is indeed the case for a square. I suppose that's intuitively clear but there are also some quantitative estimates that might be helpful more generally. The probability that the n+1-st point is contained in the convex hull should be 1 minus the expected value of the area of the n-th convex hull (for unit area regions) mathoverflow.net/questions/93099/… Aug 25 '20 at 0:03
• Hello, I meant absolutely continuous w.r.t. the Lebesgue measure, i.e. having a density. Aug 25 '20 at 0:03
• Thank you @Gabe K, I found some related results for uniform or normal distributions, and your link indeed seems to show it for a uniform distribution on a square; I was however curious about more general results for arbitrary continuous distributions Aug 25 '20 at 0:12
• Yes, if the distribution has density, you are fine. The reason is pretty straightforward: with probability $1$, the probability that a fixed $X_0$ is in the convex hull of $X_{m+1},\dots,X_{m+d+1}$ is positive for every $m$ (if $f$ is the density, that is true for every Lebesgue point of $f$, say). Thus, the probability $p(X_0,n)$ that $X_0$ is not in the convex hull of $X_1,\dots,X_n$ goes to $0$ and the dominated convergence theorem finishes the story. The interesting question is the speed of convergence, but then we need some restrictions on the density to make it meaningful. Aug 25 '20 at 0:19